University of Florida/Egm6321/f12.team5.R1.6

Show that $$ c_3(Y^1,t) \ddot Y^1 $$ is nonlinear with respect to $$ Y^1 $$

Given Equation
Equation (2) from p.5-4 gives the following expansion for the equation of motion of the wheel/magnet

It is assumed that the term $$\displaystyle u^2_{,SS}(Y^1,t)$$ is linear.

Solution
 Solved without any assistance from previous reports

From equation 1.6.1 we get, For an operator or a function to be linear, it has to satisfy the following condition:

This condition can be broken down into two separate conditions which have to be satisfied simultaneously,

1. The condition of homogeneity:

2. The condition of linearity

As both of these conditions have to be satisfied simultaneously, an operator or function that does not satisfy any one of the two conditions above can be proved as nonlinear.

Initially, checking the condition of homogeneity (Equation 1.6.4)

Now substituting $$ \alpha Y^1 $$ for $$ Y^1 $$.

Since the term $$\displaystyle u^2_{,SS}(Y^1,t)$$ is linear, we can write:

But if the term is to be homogenous then,

It is evident from Equations (1.6.7) and (1.6.8) that,

Thus, So the given term $$\displaystyle c_3(Y^1,t)\ddot Y^1$$ is not homogenous with respect to $$\displaystyle Y^1$$. As it is one of the two conditions to be simultaneously satisfied for linearity, we can say that term $$\displaystyle c_3(Y^1,t)\ddot Y^1$$ is also not linear with respect to $$\displaystyle Y^1$$.